Question

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- Answer the following questions in details. 1) Determine whether the following signals are periodic or non-periodic. If they are periodic, find the fundamental period. a) b) te=cos(+1) 2) Find the even and odd parts of the following signals: x(t) = (1 + r) cos (104) X(t) = ejt 3) A discrete-time signal [n] is shown below. Sketch and label each of the following signals. (a) xn-21 (b) x[21] (c)--) (d) x[-n21 a) 4) Determine the total energy of the following signals a) b) x[n] = u[n] x(t) = u(t), a > 0 1 system is: 5) The discrete-time system shown below is known as the unit delay element. Determine whether the (a) memoryless (b) causal (c) linear (e) stable (d) time-invariant xini Yin - xn-1) Unit delay

Answer 1

Qusty Given ů 2011) = Cos Coulet + TT 4 Hire WEI TE 271 - 271 So giura Q is signal period periodic signal nal with fundame time amental TE 27T Ans [0] = Cos IT n Bnow. Cos 20 = 2 cos'o- Cos²o 1 + cos20 2 arra this> an) - 1+ Cos 2 2" Hire Q= = TT 2 210) = +1 Cos 2 2 Ns 27T 4. - 271 x4 T nal is signal n 80 giun signali period with Time N=8 Ans 27 @). Giuco Oct) = (1+2°) Cos (lot) 3 euin pare a (t) - Xelt) = X(t) + X(t) Xell) 2 (1++3) Cosº (104) + (1-1) Cos (-10) 2 (Cos (0) = cose) Kelt) = (1++3) Cos(100) +(1-tº) Cos(dou 2

Zelt) - - Cos (104) {1+tº+1-4) 2 Cos' par! (100) 2 cos? (10) 2 Helt) = eucn - Ans odd part 9 x1t) X(t) = 21t) X(-+) 2 Xolt) - Cos (104) (1 + r) - I-t 3 Cos (10€) 2 30 (t) Cos’(10t) (+4-1+tº) Odd paro = Xolt) 2 Cos (104).t 3 Ans B) it 201t) = e. euco part = Telt) = X(t) + c(-t) 2 Cost & it -jt et e 2 Ze(t) = + jsint) + (cost-jsine) Zelt) = odd Felt) = Cost part 2 2 cost 2 X(t) - X(-+) 2 Zolt) = Ans -jt Xolt) = 2 (Cost +j sint) (Cost -jsint) 2 Zolt) = 2jsint Xolt) = Q3.7, 20i, qiuen: Q32, oc) be sint 930) Ans 3 2 - 2 0 1 2 3 4 5 6

→ it is MCCOS SI (A) x[n-2] do the “მე (lime) operation 2. sen-3) ๆ (0) by 3 n-3 4 (®) 2 [2n] it is time scaling operation with Scania factor 2(>0) 2(>1) 80 а ? al the insicot clinics 3 [20] oy Discreti siqnal 2 gioca dos ralue 2 g": (C) En]? it is time reversal operation. So WC[-n) is mirror image are [1] T[-n] 3 -4 -3 -2 n=0 (D) *C[-n +2]> it is combination of 2 and Take reuersal aduance the 3 nal by co dirse [n+2] """ 2 11 / now [-n+2] 3 3 2 111 -2 -1.

Q47, Giuin (A) oo [n] = VO] Energy of Oc[n] = 12060112 1--0 vo n7,0 Otherwise so Energy Tulni? so - 3.10 - =O U(0) The is (B) "gi D. Ans 2010) = ult) QYO X(t) Energy of E [] | olejat Ult) = ","tzo ) slevat = 0,t30 E ♡ 2at se zayde -2a 0 0 E = *(0-1) so signal 2a signal I Joule 2a 2011) = é a ult) < Energy 29 Joule Ans Q5.77 OCD) linit delay oc(n-1] 1. (a) Hire 7-གགས་པ? upon past you M= a[n-1} so oniper depend Yalue o inpu so given your will havie the a memory

B) not ality? ourpur YILI causaling so is Causal: © linearings j = 30,(n-1] and depend on Jature input yorum input X (1) then yolun yan - QX,[n-] gim 22(0-1) oyacon) = bx:(n-1) yolm Output inpur (0) - Ποω "ġ impul is and is (0) - Hire axi(n) + b T2(0) yolu ax, [n-1) + b X:(n-1] Jam = ayın) + by.cm) so_lincar. D Time inu arient? you] acIn--] [n-no-1] = [n-no-] = yen-no it is Time inuaticnt. (E) Stables you] z irongorms you, no yan-na] - So [n.no Take x[O-] 412) Ź 'x(2) 412) H(2) X(2) inuces. Žy so Here hnig ho 7th 80 finite length ={'] ot SO Think Stable Saried guanyarem ns-0 so (Memory Causal, linear, Tonu, Stable

200) 2010) is Jiuen? (A) 2010) U(1-t). uit- 0 t ult+1) (7-1) 0 >t now O t signar ate) 0 (0-4) -- 20(t) U(-+ Xu [U()- U (t-1) ] . 0 t (onton 0 (UV)-0(t-1)] now? X(t) [UL-010-)] 0 1 t © sus) 20(t) Ślt-3 2 - 80 [c(t)] 1-312 Hire XL) 8(t-t.) = X(to) (t-to) S/+-3 8(**) 2.8/t-3 2 t t=3/2 ALE) S/t-3 Am